The radius of a circle circumscribed about a regular triangle is 12√3cm. Find the perimeter of the triangle.

Let a denote the length of the side of this equilateral triangle.

Since each angle of a regular triangle is 60 °, the area S of this triangle should be equal to:

S = a * a * sin (60 °) / 2 = a ^ 2 * (√3 / 2) / 2 = a ^ 2 * (√3) / 4.

In the initial data for this task, it is reported that the radius of the circle circumscribed about this triangle is 12√3 cm.

Using the formula for the area of ​​a triangle through the radius of the circumscribed circle, we can compose the following equation:

a ^ 2 * (√3) / 4 = a ^ 3 / (4 * 12√3),

solving which, we get:

(√3) / 4 = a / (4 * 12√3);

a = (4 * 12√3 * √3) / 4 = 12 * 3 = 36 cm.

Find the perimeter of the triangle:

a + a + a = 3a = 3 * 36 = 108 cm.

Answer: 108 cm.